Find The Minimum Speed

Question

Salmon often jump waterfalls to reach their breeding grounds. One salmon starts 2.00 m from a waterfall that is 0.55 m tall and jumps at an angle of 32.0o. What must be the salmon's minimum speed to reach the waterfall?

Answer 1

Let x = the horizontal distance of travel & y = vertical component of travel t = time

v_o = initial velocity of the fish, a = angle of descent as the fish enters the jump. the x & y differential equations for the travel with gravity as the only force acting are:

y"(t) = -g and x(t)" = 0 solving with the initial conditions we get

y(t) = v_o(Sin(a)) - (g/2)*t^2 and x(t) = v_o(Cos(a))*t

Solving for t from the x equation (2.358/v_o) and placing in the y equation , we get

v_o = 6.24 m/s